Group Homomorphisms Review

Group Homomorphisms Review

We will now review some of the recent material regarding group homomorphisms.

• On the Group Homomorphisms a Homomorphism from $G$ to $H$ is a function $f : G \to H$ which has the following property that for all $x, y \in G$:
(1)
\begin{align} \quad f(x \cdot y) = f(x) * f(y) \end{align}
Theorem
(a) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $e$ is the identity in $G$ then $f(e)$ is the identity in $H$.
(b) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $x \in G$ has inverse $x^{-1} \in G$ then $f(x) \in H$ has inverse $f(x^{-1}) \in H$.
(c) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $H \subseteq G$ is a subgroup of $G$ then $f(H) \subseteq H$ is a subgroup of $H$.
(d) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $H \subseteq H$ is a subgroup of $H$ and $f^{-1}(H) \neq \emptyset$ then $f^{-1}(H)$ is a subgroup of $G$.
• On The Kernel of a Group Homomorphism page we said that if $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ where $e_2$ is the identity of $H$ then the Kernel of $f$ is defined as:
(2)
\begin{align} \quad \mathrm{ker} (f) = \{ x \in G : f(x) = e_2 \} \end{align}
• That is, $\mathrm{ker} (f)$ is the set of all elements in $G$ that are mapped to the identity element in $H$.
• We noted that $\mathrm{ker} (f)$ is a subgroup of $G$.
• We then proved an important result:
Theorem
(a) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and with identities $e_1$ and $e_2$ respectively, then $f$ is injective if and only if $\ker (f) = \{ e_1 \}$.